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Hello shubham....you have raised a very nice question...although ..few points in your discussion is quite correct and logical..but few need more light ...
# Center of mass is the point...where the whole mass of an object is assumed to be concentrated......A huge body of mass 300 Kg ..can be replaced by a point object placed at its COM of mass 300 ..to make question easier
# Center of gravity is the average location of the weight of the object...or simply what you are saying is assumed to be act ....
Now 1) If u consider more specific then this point has a bit uncertainity..but if u avoid it then it is correct
2) When gravitational field is uniform then centre of gravity coincide with COM ..hence in case when R= infinte...gravitational force become uniform in all direction...so any object placed in such situation will have it COM and COG at same location
3) In third case you are absolutely currect....because in case of extended object mass is distributed uniformaly...so instead of considering COM you need an integration to calculate Force
I hope this much ellaboration is fair enough
Cheers :D
With regards
Yagya
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